Multi-dimensional Taylor Network Optimal Output Tracking Control For Nonlinear Time-varying Systems

With the rapid development of modern science and technology and continuous expansion of production scale,the research objects become more and more complex,mainly manifested in non-linearity,time-varying,uncertainty,strong coupling,etc.Meanwhile,the requirements of automatic control system on control precision,system stability,anti-disturbance and self-adaptive ability are also getting higher and higher.Nonlinear time-varying systems contain both nonlinear terms and time-varying parameters.They are dynamic systems of general significance and widely exist in the fields of aeronautics and astronautics,industry,medicine,architecture and other disciplines.System modeling is to establish a model of the system,the relationships between the variables and parameters of the system are described according to the research needs to reflect the behavior of the actual system or some of its properties,and it is of basic importance in the process of system control and optimization.However,the nonlinear time-varying system has complex nonlinear time-varying dynamic characteristics,and the structure of the model is difficult to be determined because of lack of prior knowledge,so it is difficult to model its mechanism,especially the fast time-varying nonlinear systems.At present,the identification and control of nonlinear time-varying systems has become one of the hot and difficult problems in system science and control science.Since the development of control theory,the control theory of linear systems has matured.However,the analysis and synthesis of nonlinear control systems are much more complicated,and most of the existing control methods are based on specific types of nonlinear systems and often cannot meet the requirements of complex industrial processes.Therefore,it is necessary to develop new control methods for the analysis and synthesis of nonlinear systems.Multi-dimensional Taylor network is a new type of network structure proposed in recent years.It has simple structure,fast convergence speed,parallel processing,autonomous learning and strong nonlinear mapping abilities.It is a black box model,and has been widely used in the fields of nonlinear time series modeling and prediction,system identification,and system control.It has important theoretical significance and practical application value for the identification and control research of nonlinear time-varying systems.In this dissertation,the application of multi-dimensional Taylor network in identification and control of nonlinear time-varying systems is studied.The network structures of multi-dimensional Taylor network as identifier and controller are given respectively.An improved parameter online learning algorithm is proposed to realize online identification and real-time control of the system.The initial weights of the identifier and controller are given to improve the convergence speed and control performance.Specifically,the main research work of this dissertation is as follows:1.A multi-dimensional Taylor network optimal output tracking control scheme is proposed for a single-input single-output(SISO)nonlinear time-varying discrete system,which realizes the real-time output tracking control of the system relative to a given reference signal.Firstly,an ideal output signal is selected,the numerical solution of the optimal control law of the system relative to the ideal output signal is calculated by applying the Pontryagin minimum principle,and the corresponding optimal output is taken as the desired output signal.Secondly,a multi-dimensional Taylor network optimal controller(MTNC)is designed to fit the numerical solution of the optimal control law obtained above,and the conjugate gradient method is used to carry out off-line learning on the MTNC weights to obtain its initial weights for on-line learning.Thirdly,considering the time-varying characteristics of the system and the uncertainty of the external environment,a four-term back propagation(FBP)algorithm including the second-order momentum term and error proportional term is proposed to adjust the weight coefficients of MTNC online to realize the real-time output tracking control of the system relative to the given reference signal.Finally,the stability of FBP algorithm is analyzed,and the necessary and sufficient conditions to make it stable are determined and proved.Simulation results show the effectiveness of MTN optimal control scheme.2.Aiming at SISO nonlinear time-varying discrete system with unknown mechanism model,an optimal output tracking control scheme based on multi-dimensional Taylor network is proposed.The scheme consists of two multi-dimensional Taylor networks,one as identifier and the other as controller.Firstly,a multi-dimensional Taylor network identifier(MTNI)is constructed to establish an offline model of the system,and a set of initial MTNI weights for online identification of the system are obtained.Secondly,with respect to a given reference signal,an ideal output signal is selected.Based on the system offline identification model,the numerical solution of its optimal control law with respect to the ideal output signal is calculated by using the Pontryagin minimum principle,and its corresponding optimal output is taken as the desired output signal.Thirdly,the multi-dimensional Taylor network optimal controller(MTNC)is designed to fit the numerical solution of the optimal control law,and the conjugate gradient method is used to learn MTNC offline.The initial weights of MTNC for online learning are obtained.Finally,an improved parameter learning algorithm is proposed to adjust the weights of the identifier and the controller online respectively.The monotonic decreasing property of the error function sequences or at most the increasing property of the ? adjacent domain are analyzed and proved,which guarantees the good convergence performance of the learning algorithm.Simulation results show the effectiveness of MTN optimal control scheme.3.An optimal output tracking control scheme based on multi-dimensional Taylor network is proposed for MIMO nonlinear time-varying discrete systems.Firstly,a set of ideal output sequences is selected relative to a given reference sequence,and the numerical solution of the optimal control sequence of the system relative to the ideal output sequence is calculated by using Pontryagin minimum principle,and the corresponding optimal output sequence is taken as the desired output sequence.Secondly,multi-dimensional Taylor network optimal controller(MTNC)is designed to fit the numerical solution of the above optimal control sequence,and the conjugate gradient method is used to train MTNC weights offline to obtain its initial weights for online training.Finally,considering the time-varying characteristics of the system and the uncertainty of external environment,the back propagation(BP)algorithm is applied to update the weight of MTNC on-line to realize the real-time output tracking control of the system for the given reference sequence,and the sufficient conditions for the stability of the closed-loop system are determined and proved.Simulation results show the effectiveness of MTN optimal control scheme.4.Aiming at MIMO nonlinear fast time-varying discrete systems with unknown mechanism model,an optimal output tracking control scheme based on multi-dimensional Taylor network is proposed.The scheme consists of two multi-dimensional Taylor networks,one as identifier and the other as controller.Firstly,the multi-dimensional Taylor network identifier is constructed to establish an offline model of the system and obtain a set of initial weights for online learning of the identifier MTNI.Then,relative to a given reference sequence,a set of ideal output sequences is selected.Based on the system identification model MTNI,the numerical solution of the optimal control law of MTNI relative to the ideal output sequences is calculated by using Pontryagin minimum principle,and the corresponding optimal output sequence is taken as the desired output sequence.Thirdly,multi-dimensional Taylor network optimal controller is designed to fit the numerical solution of the above optimal control sequence,and the conjugate gradient method is applied to train MTNC weights offline to obtain the initial weight of MTNC for on-line learning.Thirdly,an improved normalized Widrow-Hoff learning algorithm is proposed to adjust the weights of the identifier online,and BP algorithm is used to train the weights of the controller online to obtain the real-time output control sequences.Finally,the convergence of the improved normalized Widrow-Hoff learning algorithm is proved,and the sufficient conditions for stabilizing the closed-loop system are determined and proved.The simulation results show the effectiveness of MTN optimal control scheme.